A difficulty in just starting a research, is that IF it is hard to even know -- what it means to have property X, such as spacetime in physics? The question of avoiding circular thought naturally appears.
To avoid circularly, we suggest to start lower than the level of measurements, in the case of spacetime or property X, before a metric function is even introduced.
We then start as a simple topological space, divisible in types by type theory only, a notion lower than sets. In that space, the metric is not introduced ad hoc, but by requiring observable properties, on the metric function.
The method, to be described in property X, mutatis mutandis as done in spacetime:
For example, because one includes as a special case the condition (2), when all such observers agree on the free speed of light (free as in vacuo), an experimental, not disputable, fact, the determination of the interval ds2 is fixed by nature as the arbiter, by physics, in spacetime.
Other arbiters are possible, such as cosmology, the mind, mathenatics, or a historically-based sequence.
In physics, this produces the only answer possible in nature for ds2, the expression for the correct metric to use, which provides the fusion of space and time in the interval ds2 -- as already known by Minkowski and Einstein, more than 100 years ago. In cosmology, with the Hubble flow, other answers are possible. The introduction of dark matter could be done this way.
The Lorentz Transformation is then, first, introduced as a consequence of this precedent, not as an axiom anymore, not before.
This simple, proper, sequence of steps, exemplified for property X as spacetime, avoids the inconsistencies of the original treatment by Einstein, and was later adopted by Einstein in formulating general relativity, as a curvature of the same spacetime.
As well-known, Shannon proposed a method for applying a mathematical form of logic called Boolean algebra to the design of relay switching circuits, creating the basis for his Information Theory (IT). This was his Master thesis and the start of our digital future.
Less well-known, it did not work well in practice. Aging batteries, noise, and speed, limited those circuits, by the two-state nature of IT, Shannon's creation. We needed tri-state, soon made possible by Verilog, an IEEE Standard, and success was reached. Noise was part of the model Itself.
Quantum theory, however, has stayed two-state, and is therefore, limited by... noise. For a tri-state quantum future, see
https://www.researchgate.net/publication/349025552_PRESENTATION_Tri-state_Quantum_Information_Model
I explained elsewhere on the danger, such as confirmation bias, of following one's opinion, or what one can imagine -- that it is better to focus on questions than (supposed) facts or (outdated) quotes. I give an example that seems far removed, but harbors the same issue.
It comes from the theory and the practice of tri-state logic design and as taught in EE in good universities. And, in Physics, let us remind ourselves that there is a notion of right and wrong --- it it agrees with Nature, it is right; but wrong otherwise. Opinions and voting do not matter in Physics, Maths, EE, and son on.
Everyone should know this, and it can change lives, can bring in new sound ideas, did influence logic, all as explained below, in different logic levels, as well.
In RG, for example, we can use it. Some answers are YES, some NO, and some are DON'T CARE. You have three logical states, but RG gives a point for any answer!
Is RG one state? Always YES? No, it's a multi-state computation, where each answer is counted according to different values, and any answer has more than three states it can be found, but RG shows only one, a point.
Go get your point, now, carelessly -- one might think, any anwer is a point -- and a disagreement gives me more points, more answers, more traffic. But, in reality, RG is part of life and life is a school. Being an RG-whale does not make one better, it actually can disqualify -- as all one can do... in life, evaluation follows several logic states, not just Boolean YES and NO.
Same with a tri-state IC. The final state of each chip IC is not just YES and NO, it can also be X. And all is computed in tri-state, one can't mix-in two-state chips, and just imagine that the end is somehow tri-state because the last chip is tri-state.
If you are paying attention, the array of tri-state chips can connect to another tristate device, compatibly. In EE, one can use even 12 states, or more, consult the IEEE for information and standards.
Silicon Valley would not exist otherwise, literally, and tri-state is a TM. It changes lives, and give sound ideas one can follow, and trade-mark, and advance technology along a more sure path -- that Physics also follows -- that of the unknown.
Boolean logic is two centuries behind, cannot compete? No, it just doesn't agree with Nature, as already measured in Physics, EE, Maths, and so on. Time to catch-up, why it makes sense in Physics, for example, to follow Minkowski SR --- that it agrees with Nature more completely, and avoids "paradoxes."
As well-known, Shannon proposed a method for applying a mathematical form of logic called Boolean algebra to the design of relay switching circuits, creating the basis for his Information Theory (IT). This was his Master thesis and the start of our digital future.
Less well-known, it did not work well in practice. Aging batteries, noise, and speed, limited those circuits, by the two-state nature of IT, Shannon's creation. We needed tri-state, soon made possible by Verilog, an IEEE Standard, and success was reached. Noise was part of the model Itself.
Quantum theory, however, has stayed two-state, and is therefore, limited by... noise. For a tri-state quantum future, see
https://www.researchgate.net/publication/349025552_PRESENTATION_Tri-state_Quantum_Information_Model
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